Carrier-phonon decoupling in perovskite thermoelectrics via entropy engineering

Thermoelectrics converting heat and electricity directly attract broad attentions. To enhance the thermoelectric figure of merit, zT, one of the key points is to decouple the carrier-phonon transport. Here, we propose an entropy engineering strategy to realize the carrier-phonon decoupling in the typical SrTiO3-based perovskite thermoelectrics. By high-entropy design, the lattice thermal conductivity could be reduced nearly to the amorphous limit, 1.25 W m−1 K−1. Simultaneously, entropy engineering can tune the Ti displacement, improving the weighted mobility to 65 cm2 V−1 s−1. Such carrier-phonon decoupling behaviors enable the greatly enhanced μW/κL of ~5.2 × 103 cm3 K J−1 V−1. The measured maximum zT of 0.24 at 488 K and the estimated zT of ~0.8 at 1173 K in (Sr0.2Ba0.2Ca0.2Pb0.2La0.2)TiO3 film are among the best of n-type thermoelectric oxides. These results reveal that the entropy engineering may be a promising strategy to decouple the carrier-phonon transport and achieve higher zT in thermoelectrics.

reflected by μ W κ L is the key point to the zT enhancement 4 .Recently, many efforts have been made to decouple carrier-phonon transport.By modulation of intrinsic crystal symmetry, e.g., Pnma to Cmcm symmetry in SnSe by Pb and Cl co-doping 5 , rhombohedral to near cubic phase transition in Pb and Sb co-doped GeTe 6 , the resulting high symmetry could benefit the improvement of the Hall mobility, while the thermal conductivity was suppressed by the dopants.Additionally, through compositing effects by adding metal particles 7 , band aligning precipitates 4,8 in the matrix phase, energy barriers for carrier transport were lowered, and the phonons were simultaneously scattered.Furthermore, interface engineering, e.g., constructing PbTe-SrTe coherent interfaces in PbTe system 9 , forming Cu 2 Se-BiCuSeO-graphene interfaces in Cu 2 Se materials 10 , was also proved effective to concurrently achieve phonon blocking and charge transmitting 9 .Despite such efforts in alloys to decouple carriers and phonons effectively, carrier-phonon decoupling is still a big challenge in thermoelectric oxides.
Oxides of low cost, low pollution, high abundance, and excellent thermal stability are competitive in thermoelectrics 11 .SrTiO 3 , as a typical perovskite thermoelectric oxide, displays relatively good electrical properties 12 (PF = S 2 σ over 1000 μW m −1 K −2 ) originating from special TiO 6 octahedrons and high symmetry cubic phase 13 , but suffers from high thermal conductivity over 11 W m −1 K −1 14 .Strategies such as nanostructuring 15 , element doping 16 , and vacancy modulation 17 , were applied in reducing lattice thermal conductivity, but the carrier mobility was sacrificed, limiting the further enhancement of zT in SrTiO 3 -based thermoelectrics 18 .Therefore, carrier-phonon decoupling is important for zT enhancement in SrTiO 3 -based oxides.Modifying the grain boundaries of SrTiO 3 -based ceramics with carbon-based materials could synergistically tune the electrical-thermal transport from the extrinsic compositing aspect [19][20][21][22][23][24] .However, it is not quite common to decouple carrier-phonon transport in SrTiO 3 by intrinsically tuning the composition and structure.
Recently, the high-entropy strategy has been found to remarkably suppress the lattice thermal conductivity in thermal barrier coatings and thermoelectrics [25][26][27] .Normally, ion displacement is strongly related to carrier transport 28 , which could be regulated in perovskitesbased dielectric capacitors by engineering entropy 29,30 .Herein, we propose that rational entropy engineering could tune the phonon and carrier transport behaviors, which could be expected to decouple the carrier-phonon transport, reaching an overall optimization of the thermoelectric performance in SrTiO 3 -based oxides.

Design and preparation of entropy-engineered thermoelectrics
As for the crystal sites for entropy engineering, since the Ti 3d orbitals form the conduction band minimum (CBM) deciding the electron transport in n-type SrTiO 3 -based semiconductors 31 , and A-site vibrations correspond to acoustic phonon branches and low-frequency optical branches dominating lattice thermal conductivity 32,33 , the A-site entropy engineering could help maintain the TiO 6 octahedrons undisturbed to prevent mobility deterioration and meanwhile strongly scatter phonons [34][35][36] , realizing carrier-phonon decoupling.Therefore, SrTiO 3 , BaTiO 3 , CaTiO 3 , and PbTiO 3 were selected to form solid solutions.Starting from La-doped SrTiO 3 and keeping the doping level (20%) of La fixed, the entropy was engineered by introducing Ba, Ca, and Pb, equimolarly occupying the rest 80% of A sites.(Sr 0.8 La 0.2 )TiO 3 (SLTO), (Sr 0.4 Ba 0.4 La 0.2 )TiO 3 (SBLTO), (Sr 0.267 Ba 0.267 Ca 0.267 La 0.2 )TiO 3 (SBCLTO), and (Sr 0.2 Ba 0.2 Ca 0.2 Pb 0.2 La 0.2 )TiO 3 (SBCPLTO) films were grown on single crystal (LaAlO 3 ) 0.3 (Sr 2 TaAlO 6 ) 0.7 (LSAT) (001) substrates by pulsed laser deposition (PLD) followed by annealing in reducing atmosphere to create O vacancies and provide electrons to make the films conductors (see below in "Methods" section).The reason for fabricating the thin films, rather than bulks, is that the highquality thin films could reduce the influence of grain boundaries in perovskites on electrical and thermal transport 37,38 , making it easier to construct entropy-transport relationship and enhance the thermoelectric properties.According to the definition of the configuration entropy S conf ig: = À Rðð , where R, N, M, x are ideal gas constant, the number of cation-site elements, the number of anion-site elements, and the molar ratio of elements 39 , the S conf ig: increased from low entropy (0.50R in SLTO) to medium entropy (1.05R, 1.38R in SBLTO, SBCLTO, respectively), and to high entropy (1.61R in SBCPLTO without taking O and Pb vacancies into account) (Supplementary Table 1) in this work.
The XRD patterns (Supplementary Figs.1-3) showed that perovskite thin films were epitaxially grown of high quality by PLD on LSAT (001) substrates, and there were no impure phases before and after the annealing process.The unchanged position and shape of LSAT peaks (Supplementary Fig. 3) indicated chemical endurance to reduce, avoiding electrical contribution from substrates 40 .The shifts after sequent introduction of Ba, Ca, Pb to (Sr 0.8 La 0.2 )TiO 3 were consistent with the relative size changes of introduced A-site atoms (Supplementary Fig. 1, Supplementary Table 2), illustrating the successful solid solution at A sites.The epitaxy nature could also be revealed by RSM of (103) peaks (Supplementary Fig. 4), and the peaks of the films deviated from the Q x of LSAT (103), which was a sign for inplane strain release by thickness of around 200 nm, excluding the influence which epitaxial strain could make on transport.The interface of the thin films and substrates consisted of co-vertex connected Al/ TaO 6 and TiO 6 octahedrons through ABF and HAADF images (Supplementary Figs. 5 and 9h), and the films showed good morphology in TEM (Supplementary Fig. 6), verifying coherently epitaxial thin films of high quality.The elements distributed uniformly at A sites in lowentropy, medium-entropy, and high-entropy samples, and no obvious ordered structure and distribution were viewed, which was reflected by EDS mapping in atomic resolution and micro-scale (Supplementary Figs.7-11), confirming the random and homogeneous occupation at A sites to fulfill the definition of high entropy.Unavoidable Pb volatilization during target sintering, film deposition, and annealing contributed to the considerable amount of Pb vacancies (Supplementary Table 3, the molar ratio Pb/Ti = 0.073 by EPMA, nominal V Pb /Ti = 0.127), adding to the entropy at A sites and tuning the A-site average radius.

Suppressing lattice thermal conductivity to amorphous limit
Since A-O skeleton dominates the transport of acoustic and lowfrequency optical phonons 17 , high level of La doping (20%) in SLTO primarily reduced lattice thermal conductivity from 11 W m −1 K −1 of STO single crystals 12 to 2.52 W m −1 K −1 due to size, mass, charge differences and the vacancies introduced (Fig. 1).The entropy engineering at A sites could further strongly suppress the lattice thermal conductivity κ L , and thermal conductivity of ~1.60 W m −1 K −1 at room temperature was measured by TDTR in SBCPLTO (Fig. 1a and Supplementary Fig. 26).After increasing entropy by introducing elements, the phonon mean free path l p decreased monotonically (Supplementary Table 7), and the room temperature lattice thermal conductivity of thin films was suppressed significantly from 2.52 W m −1 K −1 for SLTO to 1.25 W m −1 K −1 for SBCPLTO, approaching the amorphous limit 41 (Fig. 1b, Supplementary Fig. 12).To verify the thermal measurement of thin films, the lattice thermal conductivity of corresponding bulks was measured by LFA, and the results between TDTR and LFA, bulks and films, were close to some extent in Supplementary Fig. 24.
Intrinsic structural factors and extrinsic defect factors could explain the entropy-scattering relations.The acoustic and lowfrequency optical branches of phonons are mainly influenced by A sites in perovskites 32 , and thus for intrinsic factors, the doping at A sites with different mass, size, charge and vacancies could effectively scatter phonons 25,42 .The size disorder, described by a standard deviation, size disorder parameter δ (Supplementary Equation 1), could lead to lattice distortion, which is thought to be the main factor suppressing lattice thermal conductivity 43 .Due to the comparably large radius of Ba 2+ (Supplementary Table 2), the significantly increased size disorder had already strongly scattered phonons and lowered the l p (Fig. 1c).In addition to the size disorder coming from different radius of elements, the charge difference from La 3+ also could be seen as one kind of size fluctuation.Causing lattice distortion by the compensation for the local charge imbalance 25 , the charge disorder is usually responsible for  4) 43,99 at A sites.The phonon mean free path was measured on corresponding bulks at room temperature for reference (Supplementary Table 7).The definition of size disorder parameter δ can be found in supplementary materials (Supplementary Equation 1).The purple arrow is a guide for the eyes.d, e The DFT calculation of phonon dispersion of SLTO (d) and SBCPLTO (e).f Raman spectra of corresponding entropy-engineered bulks for reference.g-l The TEM, FFT, and GPA results of SLTO (g, h, i) and SBCPLTO (j, k, l).The dashed lines mark the interfaces between the films and substrates.The dislocation symbols are also shown in red.
the strong reduction in thermal conductivity 16,33 .The mass disorder could contribute to phonon scattering as well.As suggested by the phonon dispersion calculation in Fig. 1d, e and Supplementary Fig. 13, by introducing mass disorder to SrTiO 3 , the acoustic phonon branches, and low-frequency optical phonon branches became broader in dispersion, meaning the enhanced scattering rate of phonons 33 .Furthermore, since the A sites were nominally fully occupied, and the La 3+ , Sr 2+ , Ba 2+ , Ca 2+ , and Pb 2+ have no other smaller valence states when performing pulsed laser deposition, the A-site vacancies tended to form for charge balance in low-pressure, possibly contributing to the reduction in thermal conductivity by large size, charge and mass disorder 16,19,33 .Consistent with the size disorder parameter δ and DFT calculation (considering mass disorder), in corresponding unannealed bulks, the Raman spectra (Fig. 1f) which are the projection of optical phonon branches at zero point (G) of the Brillouin zone, displayed broadened peaks around 100 cm −1 from SLTO to SBCPLTO, corresponding to increased linewidth of optical branches decided by A-site vibrations 44 .For extrinsic defects, as illustrated by FFT and GPA process from HRTEM images (Fig. 1g−l), larger amounts of dislocations (as reported in high-entropy thermal barriers 27 ) and denser strain distribution, were found in high-entropy SBCPLTO than those in SLTO with lower entropy, functioning as phonon scatters of different scales.In addition, it is worth noting that though the entropy of SBCLTO is higher than SBLTO, the thermal capacity was not necessarily monotonically correlated with entropy, and the Ca with smaller mass could lead to larger thermal capacity (Dulong-Petit Law and Neumann-Kopp Rule 45 ), thus larger thermal conductivity in SBCLTO than that in SBLTO (Fig. 1b and Supplementary Fig. 24).

Improving the carrier mobility by tuning electron-conducting TiO 6 octahedrons
The carrier transport including electrical conductivity σ, Hall mobility μ H , and weighted mobility μ W 46 was measured and calculated (Fig. 2a-c and Supplementary Fig. 27).As the carrier concentration was in the order of ~10 21 cm −3 for the films (Supplementary Table 8), far larger than the critical concentration of degeneration 47 , all the films were degenerate semiconductors.For the absence of grain boundary scattering in epitaxial thin films, the σ-T, μ H -T, and μ W -T followed the acoustic phonon scattering dominated T −1.5 trend 48 .SLTO, as a typical thermoelectric material, displayed σ of ~1066 S cm −1 and μ H over 2.1 cm 2 V −1 s −1 at room temperature, while SBLTO, after the introduction of Ba with a much larger radius of 1.61 Å, showed deteriorated σ of ~175 S cm −1 and μ H of ~0.4 cm 2 V −1 s −1 at room temperature.Though the lattice thermal conductivity was low enough in SBLTO, but the coupled poor electrical properties harmed the thermoelectric performances.Substituting Ba by Ca, and Pb and simultaneously introducing Pb vacancies with lower radius, the weighted mobility recovered from SBLTO to SBCLTO to SBCPLTO, and SBCPLTO was comparable with SLTO in weighted mobility.The carrier mean free paths l c 48 and the deformation potentials Ξ def 5 were calculated (Supplementary Table 8), and obviously, with decreased deformation potentials, SBCPLTO obtained increased l c .The structural factors behind the recovery of carrier mobility needed exploring.
The relation between the mobility deterioration and the TiO 6 distortion was noticed.In Ti 2p XPS (Fig. 2d), there were no obvious Ti 3+ shoulder peaks near Ti 4+ in all films 49 , suggesting that Ti mainly existed as Ti 4+ .In this case, the shifts of Ti and O XPS binding energy peaks (Fig. 2d and Supplementary Fig. 16) were negatively correlated to the bond length.In terms of lattice parameters, SBLTO > SBCPLTO > SBCLTO > SLTO (Supplementary Fig. 14), while according to XPS, for Ti-O bond length, SBLTO > SBCLTO > SBCPLTO ≈ SLTO (Fig. 2d and Supplementary Fig. 16).SBCLTO had longer Ti-O bonds though smaller lattice than SBCPLTO, indicating that the TiO 6 octahedrons were distorted in SBCLTO, and SBCPLTO had similar Ti-O chemical surroundings with centrosymmetric SLTO despite different lattice parameters.PDF (pair distribution function) by synchrotron radiation is a good method to get information on coordination and bonds.The reverse Monte Carlo simulation could give the average bond length of atom pairs by fitting the pair distribution function (Fig. 2e), and the observed tolerance factors t obs = lengthðAÀOÞ ffiffi 2 p lengthðBÀOÞ were calculated (Supplementary Table 6) 13 .The deviation of t obs from 1 in SBLTO and SBCLTO could represent lowered symmetry and distorted TiO 6 octahedrons.
Furthermore, the types of TiO 6 distortion could also be inferred by spectral methods.In Raman spectra of films (Fig. 2f), compared with substrates, extra peaks from 240 to 400 cm −1 in films were considered to be multiple second-order peaks typical in perovskites, and extra peaks around 795 cm −1 in SBLTO and SBCLTO corresponded to the LO 4 mode, indicating off-center vibration of Ti in TiO 6 octahedrons 50 .The absence of LO 4 mode in SLTO and SBCPLTO could be thought of as a sign of Ti centrosymmetric vibrations, and the red-shifted and stronger LO 4 signal in SBLTO than that in SBCLTO might be from larger displacement and longer bond length.Additionally, from the second harmonic generation (SHG) mapping (Fig. 2g and Supplementary Fig. 17), the breaking of centrosymmetry by displacement could also be inferred.The polar domains were visualized in SBLTO, and the signal weakened in SBCLTO, hence hardly detectable in SBCPLTO with recovered mobility.The Ti displacement seemed to be the type of octahedron distortion accounting for the mobility deterioration.
To validate the TiO 6 distortion type, the Cs-corrected STEM was performed.The anti-direction tilting could double the unit cells, and lead to half-index diffraction in the selected area electron diffraction (SAED) 51 .In SLTO and SBCLTO (Supplementary Fig. 18), the presence of half-index diffraction points in SAED patterns along [110] direction ensured the tilting of TiO 6 , while no half-index diffraction points were obtained in SBLTO and SBCPLTO (Fig. 3b, c).The octahedron tilting could be seen in atomic resolution annular bright field (ABF) images by the displacement of oxygen ion (Supplementary Fig. 19), and the tilting in SBCLTO was stronger than that in SLTO, explaining the smaller binding energy in Ti 2p XPS of SBCLTO.In terms of B-site displacement, the displacement of Ti ions decreased from SBLTO to SBCLTO, and Ti ions hardly displaced in SBCPLTO (Fig. 3d-f).In short, the distortion types were slight TiO 6 tilting in SLTO, pure large Ti displacement in SBLTO, mixed TiO 6 tilting and moderate Ti displacement in SBCLTO, and minimized distortion in SBCPLTO, respectively (Supplementary Table 10).As can be seen, the TiO 6 octahedrons in SLTO are tilted, and in SBCPLTO, the tilting is ignorable, but the mobility was close between SLTO and SBCPLTO.Furthermore, the TiO 6 tilting is ignorable in SBLTO, but the mobility deteriorates significantly, which seems that the tilting is not the main factor affecting carrier mobility.The Ti displacement dominated the electron transport in our entropyengineered perovskite titanates, and with an increasing displacement of Ti, the weighted mobility decreased (Fig. 3d-g).To explain, the displaced ions could create shorter and longer bonds, and the electrons could be localized in shorter bonds with overlapping electron clouds 28 (Fig. 3a).From the aspect of orbitals, the breaking of the symmetry by Ti displacement could result in irregular bond lengths and O-Ti-O angles departing from 180°, and thus the split of t 2g orbitals would lead to carrier localization 13,52,53 .Furthermore, the inhomogeneous displacement among adjacent octahedrons in disordered medium-entropy samples could contribute to inhomogeneous electronic potentials, thus scattering electrons.After improving the symmetry, the similar bond feature could enable homogeneous electron cloud overlapping among Ti-O bonds, delocalizing the electron to transport.In summary, the Ti displacement was the distortion that hindered the electron transport in this case.

Decoupled carrier-phonon transport and enhanced thermoelectric properties
The discussion on the carrier-phonon decoupling is needed.For phonon transport, the low-entropy SLTO displayed relatively high thermal conductivity (Fig. 1a, b), and the medium-entropy SBLTO, SBCLTO, the high-entropy SBCPLTO showed decreased phonon mean free path l p and minimized thermal conductivity (Fig. 4a and Supplementary Table 7).The increase of configuration entropy (S config.) would lead to the increase in size disorder, mass disorder, and dislocation density 27 , and it could be summarized that the thermal diffusivity D decreased with increasing entropy, as shown in Fig. 4b.In terms of carrier transport, after the introduction of large Ba at A sites, the weighted mobility decreased.Interestingly, the weighted mobility was improved with suppressed phonon mean free path l p from SBLTO to SBCLTO to SBCPLTO by a decreased amount of large Ba cations and substitution by smaller Ca, Pb, and Pb vacancies (Fig. 4a).As discussed above, the displacement of Ti in TiO 6 octahedrons was the main factor affecting the carrier transport.The TiO 6 octahedrons could be tuned by changing the A-O average bond length, and the tolerance factor t = lengthðAÀOÞ ffiffi 2 p lengthðBÀOÞ was the reason behind 13 .When the tolerance factor is 1, the perovskite tends to form a cubic phase with undistorted centrosymmetric BO 6 octahedrons, and SrTiO 3 is a typical material with t ≈ 1 53 .The occupation of large Ba at A sites would deviate the t from 1, distorting the TiO 6 , and the introduction of smaller ions or vacancies to tailor t back to 1 could realize TiO 6 recovery.As concluded in Fig. 4c, with the t obs approaching 1, the TiO 6 became less distorted, performing better in weighted mobility.Since entropy is a concept of disorder, and the tolerance factor t is a mean value, it is reasonable to decouple the transport behaviors by these two different math concepts.Additionally, the entropy design was at A sites, dominating phonon transport, while the carrier transport was mainly affected by TiO 6 octahedrons, which was also the reason for the carrier-phonon decoupling.In summary, A-site rational entropy engineering resulted in A-site disorder, reducing lattice thermal conductivity, and meanwhile modulated the A-site radius and tolerance factor, thus reducing the Ti displacement and improving carrier mobility.Consequently, the carrier-phonon transport was decoupled in perovskite thermoelectrics via entropy engineering.Due to the decoupled carrier-phonon transport, the μ W /κ L increased from SBLTO to SBCPLTO, and achieved enhanced μ W /κ L of ~5.2 × 10 3 cm 3 K J −1 V −1 at room temperature in SBCPLTO, outperforming the original SLTO (Fig. 4d).
As degenerate semiconductors, the Seebeck coefficients (S) of all samples increased linearly with temperature (Supplementary Fig. 23a).High S of −228 μV K −1 was reached in SBCPLTO at 693 K, and through Pisarenko plot at room temperature using SPB model (Supplementary Fig. 23b), it could be seen that the m d * of SLTO, SBLTO and SBCLTO were close (~6.5 m 0 ), while enhanced m d * of ~8.2 m 0 was calculated in SBCPLTO.The A-site doping could help tune the energy band structure and the electrical transport in SrTiO 3 -based thermoelectrics 54,55 .To explain the improved m d *, the band dispersion was calculated by DFT (Supplementary Fig. 21), and SLTO, SBLTO, and SBCLTO shared similar band structures, while extra orbitals lay at the CBM of SBCPLTO.According to the partial DOS and COHP (Supplementary Figs.22, 23c, and 29), since Sr, Ba, and Ca belong to the same alkaline-earth main group, with empty s and d orbitals of high energy, the CBM and VBM only consist of Ti-O bonding and anti-bonding orbitals.However, Pb 2+ with lone pair 6s 2 and empty 6p could affect band structure significantly 54 , and coupled with Ti 3d and A-site d orbitals, the Pb 6p orbitals with high degeneracy could effectively contribute to the DOS at CBM to enhance the m d * of SBCPLTO.Additionally, the increased symmetry could also increase the degeneracy, thus the m d * in SBCPLTO 56 .Benefiting from recovered mobility and enhanced effective mass, the PF and electronic quality factor B E 57 of SBCPLTO outperformed the base material SLTO, and PF reached 1.1 × 10 3 μW m −1 K −2 in SBCPLTO (Fig. 4e and Supplementary Fig. 23d).In SBCPLTO, the decoupled μ W /κ L (Fig. 4f) and the increased effective mass combined helped significantly enhance the zT compared to low-and medium-entropy samples (Supplementary Fig. 23e), and the maximum zT of 0.24 was achieved at 488 K in SBCPLTO (limited by the TDTR measuring temperature).Considering the PF-T, κ L -T plateau at the middle to high-temperature range in SBCPLTO, the zT was roughly extrapolated linearly to 1173 K, reaching a high estimated zT over 0.8, which outperforms other n-type oxide thermoelectrics (Fig. 4g) 12,[58][59][60][61][62] .Due to the absence of grain boundary scattering in epitaxial thin films to enhance room temperature performances and the synergistic entropy engineering to decouple carrier-phonon transport, the average zT (RT to 473 K) of 0.14 was obtained (Supplementary Fig. 23f), which is also competitive in n-type thermoelectric oxides.

Discussion
Entropy increase usually accompanies with deteriorated carrier mobility despite lattice thermal conductivity reduction.By applying entropy engineering at phonon transport skeletons and intentionally tuning the relative sizes of ions under the guidance of factors like tolerance factor to delocalize carriers, the carrier-phonon decoupling could be realized.The strategy could be used in thermoelectrics with separate carrier and phonon transport units.Since the high entropy could suppress lattice thermal conductivity to the amorphous limit, the decoupled carrier transport could further enable competitive thermoelectric properties.This work provides solutions to applying entropy engineering in thermoelectrics and materials facing trade-offs among strongly coupled physical variables.

Sample preparation
The corresponding ceramic targets for film fabrication and the bulks for referencing characterization were fabricated by conventional solid-state reaction and pressureless sintering in air.The raw powers including La 2 O 3 powder (99.99%,Aladdin, China), TiO 2 powder (99.8%, anatase, Aladdin, China), SrCO 3 powder (99.95%,Macklin, China), BaCO 3 powder (99.95%,Macklin, China), CaCO 3 powder (99.99%,Macklin, China), PbTiO 3 powder (99.5%, Alfa Aesar, China), were stoichiometrically mixed (PbTiO 3 5% excess) by ball milling with ethanol as agent.The dried mix powders were calcined for 3 h in muffle furnaces in air (1523 K for (Sr 0.8 La 0.2 )TiO 3 , (Sr 0.4 Ba 0.4 La 0.2 )TiO 3 , (Sr 0.267 Ba 0.267 Ca 0.267 La 0.2 )TiO 3 , and 1323 K for (Sr 0.2 Ba 0.2 Ca 0.2 Pb 0.2 La 0.2 )TiO 3 ).The calcined powers were ball milled and were pressed into pellets (30 mm in diameter for targets).After cold isotropic pressing at 220 MPa, the pellets were sintered for 10 h in muffle furnaces in the air (1723 K for (Sr 0.8 La 0.2 )TiO 3 , (Sr 0.4 Ba 0.4 La 0.2 )TiO 3 , (Sr 0.27 Ba 0.27 Ca 0.27 La 0.2 )TiO 3 , and 1523K for ).The thermal diffusivity was measured on corresponding unannealed bulks at 923 K for reference.The purple arrow is a guide for the eyes.c The relation between weighted mobility and observed tolerance factor t obs. to explain the structural origin of mobility recovery.The tolerance factor was calculated from the bond length measured by PDF on corresponding bulks for reference.The purple arrow is a guide for the eyes.d The lattice thermal conductivity (κ L ), weighted mobility (μ W ), and μ W /κ L of SLTO, SBLTO, SBCLTO, and SBCPLTO to show the extent of decoupling and effects of different elements when engineering entropy at room temperature.e, f.Temperature-dependent B E (in log scale) (e), μ W /κ L (f) of entropy-engineered thin films.g Temperature-dependent zT of SBCPLTO, in comparison with other competitive n-type thermoelectric pure oxides 12,58-62,100-102 .The zT measurement was limited by the highest allowed measuring temperature of TDTR.Since the PF and κ are plateau-like with increasing T at high temperatures, the zT was extrapolated to high temperature (dashed red line).
(Sr 0.2 Ba 0.2 Ca 0.2 Pb 0.2 La 0.2 )TiO 3 ), and the dense ceramic targets were in single phases (Supplementary Fig. 25).The corresponding bulk ceramic samples for reference were prepared at the same time with targets.The corresponding bulks were not annealed to remain electrically insulated to exclude the carrier contribution to thermal transport for reference.
The thin films were grown with nominal contents (Sr 0.8 La 0.2 )TiO 3 , (Sr 0.4 Ba 0.4 La 0.2 )TiO 3 , (Sr 0.27 Ba 0.27 Ca 0.27 La 0.2 )TiO 3 , (Sr 0.2 Ba 0.2 Ca 0.2 Pb 0.2 La 0.2 )TiO 3 via pulsed laser deposition (PLD).The single crystal LSAT (001) substrates were used (12 × 4 × 0.5 mm 3 , Hefei Kejing Materials Technology Co., Ltd., China), and the high-temperature silver paste (05001-AB, 0.5 oz, Spi Supplies, the U.S.) was applied to ensure uniform temperature when depositing films.The PLD was performed with a KrF laser (wavelength = 248 nm), and the energy density, repetition rate, number of pulses, growth temperature, working oxygen partial pressure were 0.8 J cm −2 , 5 Hz, 6000 pulses, 1023 K, 20 Pa, respectively.After deposition, the films were annealed in situ in oxygen (~600 Pa) for 20 min to improve the crystal quality and were cooled down to room temperature at a rate of 10 K min −1 .The as-grown thin films underwent annealing in 5% H 2 /95% Ar mixed gas at 1173 K for 2 h to be reduced.After the annealing in reducing gas, the thin films turned into conductors, while the LSAT substrates which are not sensitive to oxygen vacancies, were tested to remain insulated, excluding the electrical contribution from the substrates 40 .

Measurement of transport behaviors
Electrical transport.The temperature-dependent electrical conductivity and Seebeck measurement of films were performed on the commercial apparatus MRS-3 (JouleYacht, China).The temperaturedependent carrier concentration (n H ) and the Hall mobility (μ H ) were calculated from σ H (μ H = σ H |R H |) and R H (n H = 1/eR H ) by van der Pauw method in Hall measurement (8408, Lake Shore Cryotronics, Inc.).The effective mass m* could be derived by combining Hall data with the Seebeck coefficient.In this work, the mobility of samples varies from 0.1 to 2.2 cm 2 V −1 s −1 , right at the test limit of DC field measurement (1-1 × 10 6 cm 2 V −1 s −1 ).Therefore, the AC field measurement developed by the Lake Shore Cryotronics 63,64 , using AC magnetic field rather than the traditional DC mode, was applied to remove the effect of misalignment, to enlarge the measuring range of Hall mobility (1 × 10 −3 to 1 × 10 6 cm 2 V −1 s −1 ) and to make the measurement more precise.According to the manual given by the Lake Shore official website, all field values in Hall measurement were nominal and could vary ±1%", and the error of μ H could be calculated by the sum of the error (1%) of five field-related variants (V H , I, B, I', U') to be 5%.The μ H -T correlation with an error of 5% was plotted in Supplementary Fig. 27a.However, multiple factors could enlarge the error of Hall measurement, like measuring geometry, electrical contact, shape and size of metal pads, etc., and the error could even reach 20%.μ H -T correlation with exaggerated errors of 20%, and 30% were plotted in Supplementary Figs.27b and 27c.As can be seen in Supplementary Fig. 27, the μ H differences between samples are still more significant than the error bar.
Thermal transport.The thermal properties of annealed LSAT(001) and annealed films were measured by the time-domain thermoreflectance (TDTR) technique 65,66 .TDTR utilizes a pump-probe configuration, where the pump beam heats the sample, and the probe beam monitors the transient sample surface temperature changes via thermoreflectance.The thermal properties are extracted by fitting the thermoreflectance signal with a thermal diffusion model.In this work, the pump beam was modulated with a frequency of 10.7 MHz, and focused on the sample surface together with the probe beam through a 10× objective lens, with a spot radius of 5.2 μm.Prior to the TDTR measurements, a 151 nm aluminum (Al) film was deposited on the film.The Al transducer layer thickness was determined by X-ray reflectance (XRR), and the thickness of the perovskite oxide film was determined by cross-sectional SEM images.The surface of Al film could be oxidized with a native oxide layer (Al 2 O 3 ) of 2-3 nm.The native oxide layer is commonly counted as an extra Al thickness of ~3 nm in other groups 67,68 .The 3 nm was added to the measured Al thickness (154 nm in total in this work), and the total thickness was used in the fitting process of the TDTR method.During the temperature-dependent thermal conductivity TDTR measurement, the chamber was vacuumed, and it could be assumed that the native oxide layer would not be further oxidized during high-temperature measurement.The approximation of adding 3 nm to the Al layer still worked in the measurement.The heat capacities of the Al transducer and the annealed LSAT (001) substrate were taken from literature value and laser flash analysis (LFA), respectively, and the heat capacities of perovskite oxide films were estimated by DSC and LFA.The thermal conductivity of the annealed LSAT substrate was measured separately using LFA (κ = C p ρD).It is important to clarify that the electrical transport in this work was measured in plane, while the thermal conductivity measured by the TDTR method was out of plane due to difficulty in technique.Additionally, the temperature range of the TDTR measurement was up to 473 K limited by the apparatus.Fortunately, for films whose thicknesses were ~200 nm, the mean free paths of phonons in entropyengineered oxides were smaller by orders (in 10 0 Å), leading to little influence from transport confinement by thickness 25,69 .And for cubic or pseudo-cubic phases, the 2nd-order tensor-like thermal conductivity could be isotropic in the strain-released epitaxial oxides with few grain boundaries 70 , the zT could be roughly calculated despite measuring in different directions.Since the PF and the thermal conductivity displayed plateau-like behaviors with increasing temperature, the zT value could be estimated by fitting the data linearly to high temperature.The laser flash apparatus (LFA, LFA457, NETZSCH, Germany) measured the thermal diffusivity and the thermal capacity of substrates and corresponding ceramics for reference, and standard samples were measured simultaneously to ensure validity.The transverse and longitudinal phonon group velocity of unannealed corresponding oxide ceramics were measured by ultrasonic pulse-echo method (Olympus 5072PR, Japan).

Characterizations
HAADF-STEM and EDS and diffraction.The cross-sectional scanning transmission electron microscope (STEM) samples were acquired by focus ion beam (FIB) milling.The samples were thinned to 50 nm by FIB with a current of 240-50 pA under an accelerating voltage of 30 kV, and with 20 pA and 5 kV during polishing process.High angle annular dark field (HAADF), annular bright field (ABF), and energy dispersion spectroscopy (EDS) images along <110> zone axis with a high resolution of 0.059 nm were collected by a 200 kV JEOL ARM 200CF equipped with double aberration correctors.The convergence angle and the collection angle were set at 25 mrad, 48-2000 mrad, respectively.The exposure time when collecting EDS signals was 2 μs, and the total time was 90 min.The noise in ABF images was filtered by HREM-Filters released by HREM Research Inc.
X-ray total scattering experiment.X-ray diffraction measurements were made at beamline 3W1 of the Beijing Synchrotron Radiation (Beijing, China) using an incident X-ray beam of wavelength 0.206468 Å (60.05 keV).A Mercu 1717HS detector (2048 × 2048 pixels of a 140 × 140-µm CsI scintillator) was placed ~150 mm downstream of the sample, giving a Q max of 25 Å −1 .The setup was calibrated using the diffraction pattern from polycrystalline CeO 2 powder.The unannealed ceramic bulks (Sr 0.8 La 0.2 )TiO 3 , (Sr 0.4 Ba 0.4 La 0.2 )TiO 3 , (Sr 0.27 Ba 0.27 Ca 0.27 La 0.2 )TiO 3 , (Sr 0.2 Ba 0.2 Ca 0.2 Pb 0.2 La 0.2 )TiO 3 were glued with Compton tapes to an aluminum alloy frame.The measurement procedure was controlled by iDetector software and 20 s exposure time was set for (Sr 0.4 Ba 0.4 La 0.2 )TiO 3 , (Sr 0.27 Ba 0.27 Ca 0.27 La 0.2 )TiO 3 , and (Sr 0.2 Ba 0.2 Ca 0.2 Pb 0.2 La 0.2 )TiO 3 , 10 s for (Sr 0.8 La 0.2 )TiO 3 due to its stronger signal.Background patterns were collected with the same setup and exposure time.The raw diffraction data were reduced from two-dimensional images and corrected for the effects of polarization and geometry using the program Fit2D 71 and absorption, geometry, detector effects, and the normalization procedure carried out using PDFgetX2 72 .
Reverse Monte Carlo (RMC) simulation.Using RMC simulation, the structural models were derived from the diffraction data 73 .To start, the configuration of ~5000 atoms was used.Constraints were set, including the connectivity of Ti-O bonds (i.e. Ti was coordinated to a reasonable number of O up to 2.5 Å) and the atom-atom approaches.The aim of setting the constraints was to avoid the results of unrealistic structures in physics.By counting the atomic configurations generated by RMC simulations, the structural information was derived.
SHG. Second harmonic generation (SHG) spectroscopy was performed under a home-designed SHG microscope 74 .The excitation laser was provided by a Ti: sapphire mode-locking femtosecond laser (MaiTai SP, Spectra-Physics) which generated 35 fs pulses with a repetition rate of 80 MHz and a wavelength centered at 800 nm.The fundamental laser beam was directed onto the sample at normal incidence and focused to a focal spot diameter of ≈1 μm using a 50× (numerical aperture NA = 0.55) objective.The back-scattered SHG signal was collected with the same objective and detected using a Hamamatsu photomultiplier tube.The SHG microscopy images of the square areas selected were acquired by scanning the probing position using a pair of galvanometers.When scanning the interested region, the angle between the polarization direction of the incident light and the SHG light collected was fixed the same at 90°.The rate of scanning was ~50 ms pixel −1 and ~2 μm s −1 .

Fig. 1 |
Fig. 1 | Thermal transport behaviors of entropy-engineered thin films, and the extrinsic and intrinsic origins of strong phonon scattering after entropy increase.a Temperature-dependent total thermal conductivity (κ) of entropyengineered perovskite oxide thin films.b Calculated lattice thermal conductivity (κ L ) of entropy-engineered thin films and amorphous limit thermal conductivity of SBCPLTO.c The relation between phonon mean free path (l p ) and the size disorder parameter (δ) (Supplementary Table4)43,99 at A sites.The phonon mean free path was measured on corresponding bulks at room temperature for reference

_Fig. 2 |
Fig. 2 | Electrical transport behaviors of entropy-engineered thin films and the possible structural origins of decoupled carrier transport explained by spectral methods.a-c Temperature-dependent electrical conductivity (σ) (a), Hall mobility (μ H ) (b), and weighted mobility (μ W ) (c) of entropy-engineered thin films.The acoustic phonon scattering dominated T −1.5 trend curves were plotted (dashed gray line).d The XPS Ti 2p spectra of entropy-engineered thin films.The dashed gray line marks the peak position and serves as a guide for the eyes.e The PDF results of corresponding bulks to get referencing bond length and the fitting curves by reverse Monte Carlo methods.f The Raman spectra of LSAT substrates and entropy-engineered thin films on LSAT substrates.The typical vibration modes of perovskite thin films were highlighted by dashed lines, and the inset is the schematic of LO 4 vibration mode.g SHG mapping of SBLTO, SBCLTO, and SBCPLTO in 10 μm × 10 μm.

Fig. 3 |
Fig. 3 | Atomic scale electron microscopy characterization to explain the recovered mobility of entropy-engineered thin films.a Schematic of the possible mechanism of electron scattering and localization in Ti-displaced TiO 6 octahedrons.C point marked is the center of the TiO 6 octahedron, and the length of Ti displacement is the length of C-Ti.b, c SAED patterns along [110] to prove the absence of octahedron tilting in SBLTO and SBCPLTO.d-f ABF images in the atomic resolution after filtering to show the Ti displacement in SBLTO (d), SBCLTO (e), and SBCPLTO (f).The orange scale bar denotes 5 Å.The orange, red, and blue spheres represent A-site atoms, O, and Ti, and the overlapped spheres were omitted.g The relation between room temperature weighted mobility (μ W ) and average normalized Ti displacement ( d Ti ) of SBLTO, SBCLTO, and SBCPLTO with increased entropy.The displacement was normalized by the ratio d Ti = length(C-Ti)/length(C-O b ) in (a).The error bar is the standard deviation δ d .The blue and red arrows are the guides to show the trends of weighted mobility and displacement.The details of the analysis and results can be found in supplementary materials.

Fig. 4 |
Fig. 4 | Decoupled carrier-phonon transport and enhanced thermoelectric properties of entropy-engineered thin films.a Schematic diagram of tuning A sites and TiO 6 octahedrons to decouple the carrier-phonon transport with increasing entropy.The symbols of elements were shown in the insets.b The correlation between the thermal diffusivity (D) and the nominal configuration entropy (S config.).The thermal diffusivity was measured on corresponding unannealed bulks at 923 K for reference.The purple arrow is a guide for the eyes.c The relation between weighted mobility and observed tolerance factor t obs. to explain the structural origin of mobility recovery.The tolerance factor was calculated from the bond length measured by PDF on corresponding bulks for reference.The